Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Fokker-Planck equation for a test-particle weakly coupled to a magnetized plasma

We consider the derivation of a kinetic equation for a charged test particle weakly interacting with an electrostatic plasma in thermal equilibrium, subject to a uniform external magnetic field. The Liouville equation leads to a generalized master equation to second order in the `weak' interaction; a Fokker-Planck-type equation then follows as a `Markovian' approximation. It is shown that such an equation does not preserve the positivity of the distribution function f(x,v;t). By applying techniques developed in the theory of open systems, a correct Fokker-Planck equation is derived. Explicit expressions for the diffusion and drift coefficients, depending on the magnetic field, are obtained.

Solutions of the Fokker-Planck equation for a Morse isospectral potential

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A General Solution of the Fokker-Planck Equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Fokker-Planck Equation for a Metastable Time Dependent Potential

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The Fokker-Planck equation for a bistable potential

The Fokker-Planck equation is studied through its relation to a Schrodinger-type equation. The advantage of this combination is that we can construct the probability distribution of the Fokker-Planck equation by using well-known solutions of the Schrodinger equation. By making use of such a combination, we present the solution of the Fokker-Planck equation for a bistable potential related to a double oscillator. Thus, we can observe the temporal evolution of the system describing its dynamic properties such as the time tau to overcome the barrier. By calculating the rates k = 1/tau as a function of the inverse scaled temperature 1/D, where D is the diffusion coefficient, we compare the aspect of the curve k x 1/D, with the ones obtained from other studies related to four different kinds of activated process. We notice that there are similarities in some ranges of the scaled temperatures, where the different processes follow the Arrhenius behavior. We propose that the type of bistable potential used in this study may be used, qualitatively, as a simple model, whose rates share common features with the rates of some single rate-limited thermally activated processes. (C) 2014 Elsevier B.V. All rights reserved.

Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems

We derive rigorously the Fokker-Planck equation that governs the statistics of soliton parameters in optical transmission lines in the presence of additive amplifier spontaneous emission. We demonstrate that these statistics are generally non-Gaussian. We present exact marginal probability-density functions for soliton parameters for some cases. A WKB approach is applied to describe the tails of the probability-density functions. © 2005 Optical Society of America.